Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×#3289 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
834 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 834 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,17,33)(8,19,21,20)(9,23,25,13)(10,14)(11,16,26,32)(12,22)(18,29,24,34)(27,35,28,36)(30,31);(5,7,10,18)(6,11,14,28)(8,16,21,36)(9,24,25,17)(12,29,31,15)(13,26,23,27)(19,34,20,33)(22,32,30,35);(2,4)(5,6)(7,11)(8,20)(9,23)(10,14)(12,30)(13,25)(15,32)(16,33)(17,26)(18,28)(19,21)(22,31)(24,27)(29,35)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dh,e^b=eghi,f^b=fghi,g^b=gi,d^c=dgi,g^c=gi,h^c=hi,e^d=ehi,f^d=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59379);